Question

The number of ways in which $9$ persons can be divided into three equal groups is?

• A. $1680$
• B. $840$
• C. $560$
• D. $280$

Answer 1

The correct option is D

$280$

Explanation for the correct option:

Find the number of ways of required distribution.

Since, the number of ways in which $n$ objects can be distributed in $r$ groups of equal size is given by $\frac{n!}{{\left[\left({\displaystyle \frac{n}{r}}\right)!\right]}^r·r!}$.

So, the total number of ways in which $9$ persons can be divided into three equal groups is $\frac{9!}{{\left[\left({\displaystyle \frac{9}{3}}\right)!\right]}^3·3!}$.

$$\begin{gathered} \frac{9!}{{\left[\left({\displaystyle \frac{9}{3}}\right)!\right]}^3·3!}=\frac{9!}{{(3!)}^3·3!} \\ \Rightarrow \frac{9!}{{\left[\left({\displaystyle \frac{9}{3}}\right)!\right]}^3·3!}=280 \end{gathered}$$

Therefore, The number of ways in which $9$ persons can be divided into three equal groups is $280$.

Hence, option $D$ is the correct answer.